The fishnet CFT is a non-unitary CFT of two matrix complex scalar fields interacting via a single quartic potential. The chiral nature of the interaction strongly constrains the Feynman diagrams arising at each order in perturbation theory, those that survive are of fishnet type. In this talk, I will present the TBA equations for the conformal dimensions of multi-magnon local operators in this theory. I will emphasize the need to diagonalize suitable graph-building operators in order to determine the asymptotic data, dispersion relation and S matrix, on which the TBA relies. A dual version of the TBA equations, relating D-dimensional graphs to two-dimensional sigma models, will also be examined. The last part of the talk will be devoted to the presentation of the underlying non-compact spin chain and of additional results regarding diagonalization of graph-building operators. --------------------- Part of London Integrability Journal Club. New participants please register at integrability-london.weebly.com for the link.